A Hidden Signal in the Ulam Sequence

ABSTRACT The Ulam sequence is defined as a1 = 1, a2 = 2, and an being the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives Ulam remarked that understanding the sequence, which has been described as “quite erratic,” seems difficult and indeed nothing is known. We report the empirical discovery of a surprising global rigidity phenomenon: there seems to exist a real α ∼ 2.5714474995… such that supported on a subset of . Indeed, for the first 107 elements of Ulam’s sequence, The same phenomenon arises for some other initial conditions a1, a2: the distribution functions look very different from each other and have curious shapes. A similar but more subtle phenomenon seems to arise in Lagarias’ variant of MacMahon’s “primes of measurement” sequence.

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